Monthly Archives: February 2017

When one looks through the history of condensed matter experiment, it is strange to see how many times discoveries were made in a serendipitous fashion (see here for instance). I would argue that most groundbreaking findings were unanticipated. The discoveries of superconductivity by Onnes, the Meissner effect, superfluidity in He-4, cuprate (and high temperature) superconductivity, the quantum Hall effect and the fractional quantum Hall effect were all unforeseen by the very experimentalists that were conducting the experiments! Theorists also did not anticipate these results. Of course, a whole slew of phases and effects were theoretically predicted and then experimentally observed as well, such as Bose-Einstein condensation, the Kosterlitz-Thouless transition, superfluidity in He-3 and the discovery of topological insulators, not to diminish the role of prediction.

For the condensed matter experimentalist, though, this presents a rather strange paradigm. Naively (and I would say that the general public by and large shares this view), science is perceived as working within a question and answer framework. You pose a concrete question, and then conduct and experiment to try to answer said question. In condensed matter physics, this often not the case, or at least only loosely the case. There are of course experiments that have been conducted to answer concrete questions — and when they are conducted, they usually end up being beautiful experiments (see here for example). But these kinds of experiments can only be conducted when a field reaches a point where concrete questions can be formulated. For exploratory studies, the questions are often not even clear. I would, therefore, consider these kinds of Q&A experiments to be the exception to the rule rather than the norm.

More often then not, discoveries are made by exploring uncharted territory, entering a space others have not explored before, and tempting fate. Questions are often not concrete but posed in the form, “What if I do this…?”. I know that this makes condensed matter physics sound like it lacks organization, clarity and structure. But this is not totally untrue. Most progress in the history of science did not proceed in a straight line like textbooks make it seem. When weird particles were popping up all over the place in particle physics in the 1930s and 40s, it was hard to see any organizing principles. Experimentalists were discovering new particles at a rate with which theory could not keep up. Only after a large number of particles had been discovered did Gell-Mann come up with his “Eightfold Way”, which ultimately led to the Standard Model.

This is all to say that scientific progress is tortuous, thought processes of scientists are highly nonlinear, and there is a lot of intuition required in deciding what problems to solve or what space is worth exploring. In condensed matter experiment, it is therefore important to keep pushing boundaries of what has been done before, explore, and do something unique in hope of finding something new!

Exposure to a wide variety of observations and methods is required to choose what boundaries to push and where to spend one’s time exploring. This is what makes diversity and avoiding “herd thinking” important to the scientific endeavor. Exploratory science without concrete questions makes some (especially younger graduate students) feel uncomfortable, since there is always the fear of finding nothing! This means that condensed matter physics, despite its tremendous progress over the last few decades, where certain general organizing principles have been identified, is still somewhat of a “wild west” in terms of science. But it is precisely this lack of structure that makes it particularly exciting — there are still plenty of rocks that need overturning, and it’s hard to foresee what is going to be found underneath them.

In experimental science, questions are important to formulate — but the adventure towards the answer usually ends up being more important than the answer itself.

On a recent plane ride, I was able to catch an episode of the new PBS series Genius by Stephen Hawking. I was surprised by the quality of the show and in particular, its emphasis on experiment. Usually, shows like this fall into the trap of giving one the facts (or speculations) without an adequate explanation of how scientists come to such conclusions. However, this one is a little different and there is a large emphasis on experiment, which, at least to me, is much more inspirational.

There is also another cute example that illustrates why overtones of the natural harmonic frequency components result when subject to slightly nonlinear oscillations. The solution to this problem therefore shows why harmonic distortions often affect speakers; sometimes speakers emit frequencies not present in the original electrical signal. Furthermore, it shows why second harmonic generation can result when intense light is incident on a material.

First, imagine a perfectly harmonic oscillator with a potential of the form . We know that such an oscillator, if displaced from its original position, will result in oscillations at the natural frequency of the oscillator with the position varying as . The potential and the position of the oscillator as a function of time are shown below:

(Left) Harmonic potential as a function of position. (Right) Variation of the position of the oscillator with time

Now imagine that in addition to the harmonic part of the potential, we also have a small additional component such that , so that the potential now looks like so:

The equation of motion is now nonlinear:

where and are constants. It is easy to see that if the amplitude of oscillations is small enough, there will be very little difference between this case and the case of the perfectly harmonic potential.

However, if the amplitude of the oscillations gets a little larger, there will clearly be deviations from the pure sinusoid. So then what does the position of the oscillator look like as a function of time? Perhaps not too surprisingly, considering the title, is that not only are there oscillations at , but there is also an introduction of a harmonic component with .

While the differential equation can’t be solved exactly without resorting to numerical methods, that the harmonic component is introduced can be seen within the framework of perturbation theory. In this context, all we need to do is plug the solution to the simple harmonic oscillator, into the nonlinear equation above. If we do this, the last term becomes:

,

showing that we get oscillatory components at twice the natural frequency. Although this explanation is a little crude — one can already start to see why nonlinearity often leads to higher frequency harmonics.

With respect to optical second harmonic generation, there is also one important ingredient that should not be overlooked in this simplified model. This is the fact that frequency doubling is possible only when there is an component in the potential. This means that the potential needs to be inversion asymmetric. Indeed, second harmonic generation is only possible in inversion asymmetric materials (which is why ferroelectric materials are often used to produce second harmonic optical signals).

Because of its conceptual simplicity, it is often helpful to think about physical problems in terms of the classical harmonic oscillator. It would be interesting to count how many Nobel Prizes have been given out for problems that have been reduced to some variant of the harmonic oscillator!